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The Helgason Fourier transform for homogeneous vector bundles over compact Riemannian symmetric spaces – the local theory. (English) Zbl 1060.43004
From the author’s summary: The Helgason Fourier transform on noncompact Riemannian symmetric spaces $$G/K$$ is generalized to the homogeneous vector bundles over the compact dual spaces $$U/K.$$ The scalar theory on $$U/K$$ was considered by Sherman (the local theory for $$U/K$$ of arbitrary rank, and the global theory for $$U/K$$ of rank one). In this paper we extend the local theory of Sherman to arbitrary homogeneous vector bundles on $$U/K.$$ For $$U/K$$ of rank one we also obtain a generalization of the Cartan-Helgason theorem valid for any $$K$$-type.

##### MSC:
 43A85 Harmonic analysis on homogeneous spaces 42C15 General harmonic expansions, frames 22E46 Semisimple Lie groups and their representations
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##### References:
 [1] Camporesi, R., The helgason Fourier transform for homogeneous vector bundles over Riemannian symmetric spaces, Pacific J. math., 179, 263-300, (1997) · Zbl 0871.43006 [2] R. Camporesi, A generalization of the Cartan-Helgason theorem for Riemannian symmetric spaces of rank one, Pacific J. Math., to appear. · Zbl 1100.43004 [3] Helgason, S., Groups and geometric analysis, (1984), Academic Press New York [4] S. Helgason, Geometric Analysis on Symmetric Spaces, Mathematical Surveys and Monographs, vol. 39, American Mathematical Society, Providence, RI, 1994. · Zbl 0809.53057 [5] Knapp, A.W., Representation theory of semisimple groups, an overview based on examples, (1986), Princeton University Press Princeton, NJ · Zbl 0604.22001 [6] B. Kostant, A branching law for subgroups fixed by an involution and a noncompact analogue of the Borel-Weil theorem, “Noncommutative harmonic analysis”, 291-353, Progress in Mathematics, vol. 220, Birkhäuser, Boston, MA, 2004. · Zbl 1162.17304 [7] Sherman, T., The helgason Fourier transform for compact Riemannian symmetric spaces of rank one, Acta math., 164, 73-144, (1990) · Zbl 0707.43001
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